Lect12_Wavepacket - Momentum vs Wavevector • Instead of...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Momentum vs. Wavevector • Instead of momentum, it is often convenient to use wavevector states: h P K = k k k K = ) ( k k k k ′ − = ′ δ π 2 x k i e k x = 1 = ∫ + ∞ ∞ − k k dk [ ] i K X = , [ ] , = K P k k k k k k p p p p ′ = ′ − = ′ − = ′ − = ′ h h h h 1 ) ( 1 ) ( ) ( δ δ δ ∴ k = h p p = h k Wavevector definition: Important commutators: Eigenvalue equation: Relation to momentum eigenstates: k p p c k h = = Normalization Closure Wavefunction Wavefunction in K-space • What is the wavefunction in K-basis? ) ( ) ( ) , ( 2 2 2 2 k k e k k e t k t k m t k i m t k i − = = = − − δ ψ ψ h h k k ) , ( t k ψ ) ( k = ψ Let 2 2 ) ( k e t t M k i h − = ψ Then: • Quantum mechanical kinetic-energy eigenstate: – Only Global phase is changing in time – Global phase can’t be observed • Is anything actually moving? – The spatial phase-pattern is moving, but at: – This is called the `phase velocity’ • Not related to particle velocity in classical limit • Classical free-particle motion: • How can we have a `classical limit’ of QM if nothing moves at v=p /M? Wavefunction in X-space ) ( ) ( p t p t v x t x = + = 2 2 ) ( p e t t M p i h − = ψ h h π ψ 2 ) , ( 2 − = t M p x p i e t x M p v 2 = h π ψ 2 1 ) , ( 2 = t x `Motion ’ in QM • The answer is that in QM motion is in interference effect • Consider a quantum-superposition of two plane-waves: • The interference pattern can move!...
View Full Document

This note was uploaded on 12/21/2011 for the course PHY 851 taught by Professor Moore,m during the Fall '08 term at Michigan State University.

Page1 / 10

Lect12_Wavepacket - Momentum vs Wavevector • Instead of...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online